Advertisement

Boundary Value Problems

, 2006:32950 | Cite as

Radial solutions for a nonlocal boundary value problem

  • Ricardo Enguiça
  • Luís Sanchez
Open Access
Research Article

Abstract

We consider the boundary value problem for the nonlinear Poisson equation with a nonlocal term Open image in new window , Open image in new window . We prove the existence of a positive radial solution when Open image in new window grows linearly in Open image in new window , using Krasnoselskiiés fixed point theorem together with eigenvalue theory. In presence of upper and lower solutions, we consider monotone approximation to solutions.

Keywords

Differential Equation Partial Differential Equation Ordinary Differential Equation Functional Equation Point Theorem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Bebernes JW, Lacey AA: Global existence and finite-time blow-up for a class of nonlocal parabolic problems. Advances in Differential Equations 1997,2(6):927-953.MathSciNetMATHGoogle Scholar
  2. 2.
    Chang N-H, Chipot M: On some mixed boundary value problems with nonlocal diffusion. Advances in Mathematical Sciences and Applications 2004,14(1):1-24.MathSciNetGoogle Scholar
  3. 3.
    De Coster C, Habets P: The lower and upper solutions method for boundary value problems. In Handbook of Differential Equations. Elsevier, New York; North-Holland, Amsterdam; 2004:69-160.Google Scholar
  4. 4.
    Deimling K: Nonlinear Functional Analysis. Springer, Berlin; 1985:xiv+450.CrossRefMATHGoogle Scholar
  5. 5.
    Fijałkowski P, Przeradzki B: On a radial positive solution to a nonlocal elliptic equation. Topological Methods in Nonlinear Analysis 2003,21(2):293-300.MathSciNetMATHGoogle Scholar
  6. 6.
    Freitas P, Sweers G: Positivity results for a nonlocal elliptic equation. Proceedings of the Royal Society of Edinburgh. Section A. Mathematics 1998,128(4):697-715. 10.1017/S0308210500021727MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Gomes JM, Sanchez L: On a variational approach to some non-local boundary value problems. Applicable Analysis 2005,84(9):909-925. 10.1080/00036810500048202MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Gaudenzi M, Habets P, Zanolin F: Positive solutions of singular boundary value problems with indefinite weight. Bulletin of the Belgian Mathematical Society. Simon Stevin 2002,9(4):607-619.MathSciNetMATHGoogle Scholar
  9. 9.
    Jiang D, Gao W, Wan A: A monotone method for constructing extremal solutions to fourth-order periodic boundary value problems. Applied Mathematics and Computation 2002,132(2-3):411-421. 10.1016/S0096-3003(01)00201-6MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Pao CV: Nonlinear Parabolic and Elliptic Equations. Plenum Press, New York; 1992:xvi+777.MATHGoogle Scholar
  11. 11.
    Smirnov V: Cours de Mathématiques Supérieures. Volume 2. Mir, Moscoú; 1970.Google Scholar
  12. 12.
    Walter W: Ordinary Differential Equations, Graduate Texts in Mathematics. Volume 182. Springer, New York; 1998:xii+380.Google Scholar
  13. 13.
    Yang Z: Positive solutions to a system of second-order nonlocal boundary value problems. Nonlinear Analysis. Theory, Methods & Applications 2005,62(7):1251-1265. 10.1016/j.na.2005.04.030MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Zeidler E: Nonlinear Functional Analysis and Its Applications—I : Fixed-Point Theorems. Springer, New York; 1986:xxi+897.CrossRefMATHGoogle Scholar

Copyright information

© Enguiça and Sanchez 2006

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Authors and Affiliations

  1. 1.Área Científica de MatemáticaInstituto Superior de Engenharia de Lisboa, Rua Conselheiro Emídio NavarroLisboaPortugal
  2. 2.Faculdade de Ciências da Universidade de LisboaLisboaPortugal

Personalised recommendations